In this section we discuss how to go between categories cofibred in groupoids over \mathcal{C}_\Lambda to categories cofibred in groupoids over \widehat{\mathcal{C}}_\Lambda and vice versa.
Definition 90.7.1. Let \mathcal{F} be a category cofibered in groupoids over \mathcal{C}_\Lambda . The category \widehat{\mathcal{F}} of formal objects of \mathcal{F} is the category with the following objects and morphisms.
A formal object \xi = (R, \xi _ n, f_ n) of \mathcal{F} consists of an object R of \widehat{\mathcal{C}}_\Lambda , and a collection indexed by n \in \mathbf{N} of objects \xi _ n of \mathcal{F}(R/\mathfrak m_ R^ n) and morphisms f_ n : \xi _{n + 1} \to \xi _ n lying over the projection R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n.
Let \xi = (R, \xi _ n, f_ n) and \eta = (S, \eta _ n, g_ n) be formal objects of \mathcal{F}. A morphism a : \xi \to \eta of formal objects consists of a map a_0 : R \to S in \widehat{\mathcal{C}}_\Lambda and a collection a_ n : \xi _ n \to \eta _ n of morphisms of \mathcal{F} lying over R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n, such that for every n the diagram
\xymatrix{ \xi _{n + 1} \ar[r]_{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n }
commutes.
The category of formal objects comes with a functor \widehat{p}: \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda which sends an object (R, \xi _ n, f_ n) to R and a morphism (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n) to the map R \to S.
Lemma 90.7.2. Let p : \mathcal{F} \to \mathcal{C}_\Lambda be a category cofibered in groupoids. Then \widehat{p} : \widehat{\mathcal{F}} \to \widehat{\mathcal{C}}_\Lambda is a category cofibered in groupoids.
Proof.
Let R \to S be a ring map in \widehat{\mathcal{C}}_\Lambda . Let (R, \xi _ n, f_ n) be an object of \widehat{\mathcal{F}}. For each n choose a pushforward \xi _ n \to \eta _ n of \xi _ n along R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n. For each n there exists a unique morphism g_ n : \eta _{n + 1} \to \eta _ n in \mathcal{F} lying over S/\mathfrak m_ S^{n + 1} \to S/\mathfrak m_ S^ n such that
\xymatrix{ \xi _{n + 1} \ar[d] \ar[r]_{f_ n} & \xi _ n \ar[d] \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n }
commutes (by the first axiom of a category cofibred in groupoids). Hence we obtain a morphism (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n) lying over R \to S, i.e., the first axiom of a category cofibred in groupoids holds for \widehat{\mathcal{F}}. To see the second axiom suppose that we have morphisms a : (R, \xi _ n, f_ n) \to (S, \eta _ n, g_ n) and b : (R, \xi _ n, f_ n) \to (T, \theta _ n, h_ n) in \widehat{\mathcal{F}} and a morphism c_0 : S \to T in \widehat{\mathcal{C}}_\Lambda such that c_0 \circ a_0 = b_0. By the second axiom of a category cofibred in groupoids for \mathcal{F} we obtain unique maps c_ n : \eta _ n \to \theta _ n lying over S/\mathfrak m_ S^ n \to T/\mathfrak m_ T^ n such that c_ n \circ a_ n = b_ n. Setting c = (c_ n)_{n \geq 0} gives the desired morphism c : (S, \eta _ n, g_ n) \to (T, \theta _ n, h_ n) in \widehat{\mathcal{F}} (we omit the verification that h_ n \circ c_{n + 1} = c_ n \circ g_ n).
\square
Definition 90.7.3. Let p : \mathcal{F} \to \mathcal{C}_\Lambda be a category cofibered in groupoids. The category cofibered in groupoids \widehat{p} : \widehat{\mathcal F} \to \widehat{\mathcal{C}}_\Lambda is called the completion of \mathcal{F}.
If \mathcal{F} is a category cofibered in groupoids over \mathcal C_\Lambda , we have defined \widehat{\mathcal{F}}(R) for R \in \mathop{\mathrm{Ob}}\nolimits (\widehat{\mathcal{C}}_\Lambda ) in terms of the filtration of R by powers of its maximal ideal. But suppose \mathcal{I} = (I_ n) is a filtration of R by ideals inducing the \mathfrak {m}_ R-adic topology. We define \widehat{\mathcal{F}}_\mathcal {I}(R) to be the category with the following objects and morphisms:
An object is a collection (\xi _ n, f_ n)_{n \in \mathbf{N}} of objects \xi _ n of \mathcal{F}(R/I_ n) and morphisms f_ n : \xi _{n + 1} \to \xi _ n lying over the projections R/I_{n + 1} \to R/I_ n.
A morphism a : (\xi _ n, f_ n) \to (\eta _ n, g_ n) consists of a collection a_ n : \xi _ n \to \eta _ n of morphisms in \mathcal{F}(R/I_ n), such that for every n the diagram
\xymatrix{ \xi _{n + 1} \ar[r]^{f_ n} \ar[d]_{a_{n + 1}} & \xi _ n \ar[d]^{a_ n} \\ \eta _{n + 1} \ar[r]^{g_ n} & \eta _ n }
commutes.
Lemma 90.7.4. In the situation above, \widehat{\mathcal{F}}_\mathcal {I}(R) is equivalent to the category \widehat{\mathcal{F}}(R).
Proof.
An equivalence \widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R) can be defined as follows. For each n, let m(n) be the least m that I_ m \subset \mathfrak m_ R^ n. Given an object (\xi _ n, f_ n) of \widehat{\mathcal{F}}_\mathcal {I}(R), let \eta _ n be the pushforward of \xi _{m(n)} along R/I_{m(n)} \to R/\mathfrak m_ R^ n. Let g_ n : \eta _{n + 1} \to \eta _ n be the unique morphism of \mathcal{F} lying over R/\mathfrak m_ R^{n + 1} \to R/\mathfrak m_ R^ n such that
\xymatrix{ \xi _{m(n + 1)} \ar[rrr]_{f_{m(n)} \circ \ldots \circ f_{m(n + 1) - 1}} \ar[d] & & & \xi _{m(n)} \ar[d] \\ \eta _{n + 1} \ar[rrr]^{g_ n} & & & \eta _ n }
commutes (existence and uniqueness is guaranteed by the axioms of a cofibred category). The functor \widehat{\mathcal{F}}_\mathcal {I}(R) \to \widehat{\mathcal{F}}(R) sends (\xi _ n, f_ n) to (R, \eta _ n, g_ n). We omit the verification that this is indeed an equivalence of categories.
\square
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